Free Access
Volume 55, Number 3, May-June 2021
Page(s) 1005 - 1037
Published online 05 May 2021
  1. A. Abdulle, Analysis of a heterogeneous multiscale FEM for problems in elasticity. Math. Models Methods Appl. Sci. 16 (2006) 615–635. [Google Scholar]
  2. D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429–2451. [Google Scholar]
  3. D.N. Arnold, R.S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76 (2007) 1699–1723. [Google Scholar]
  4. D.N. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions. Math. Comp. 77 (2008) 1229–1251. [Google Scholar]
  5. D.N. Arnold, G. Awanou, B.W. Bestbury and W. Qiu, Mixed finite elements for elasticity on quadrilateral meshes. Adv. Comput. Math. 41 (2015) 553–572. [Google Scholar]
  6. D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8 (2009) 95–121. [MathSciNet] [Google Scholar]
  7. D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics. Springer-Verlag, New York (2013). [Google Scholar]
  8. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. RAIRO: Anal. Numér. 2 (1974) 129–151. [Google Scholar]
  9. F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems, edited by W. Hackbusch. Braunschweig, Wiesbaden (1984) 11–19. [Google Scholar]
  10. F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. [Google Scholar]
  11. M. Buck, O. Iliev and H. Andrä, Multiscale finite element coarse spaces for the application to linear elasticity. Cent. Eur. J. Math. 11 (2013) 608–701. [Google Scholar]
  12. M. Buck, O. Iliev and H. Andrä, Multiscale finite elements for linear elasticity: oscillatory boundary conditions. In Lect. Not. Comp. Sci. Springer International Publishing (2014) 237–245. [Google Scholar]
  13. B. Cockburn, J. Gopalakrishnan and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79 (2010) 1331–1349. [Google Scholar]
  14. M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM:M2AN 7 (1973) 33–75. [EDP Sciences] [Google Scholar]
  15. J. de la Puente, HPC4E Seismic Test Suite. (2016). [Google Scholar]
  16. L. Demkowicz, Polynomial exact sequences and projection-based interpolation with application to Maxwell equations. In: Mixed Finite Elements, Compatibility Conditions, and Applications, edited by D. Boffi and L. Gastaldi. Vol. 1939 of Lecture Notes in Mathematics. Springer, Berlin, Heidelberg (2008) 101–158. [Google Scholar]
  17. P.R.B. Devloo, C.M.A.A. Bravo and E.C. Rylo, Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements. Comput. Meth. Appl. Mech. Eng. 198 (2009) 1716–1725. [Google Scholar]
  18. P.R.B. Devloo, A.M. Farias and S.M. Gomes, A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations. Comput. Math. Appl. 77 (2019) 1864–1872. [Google Scholar]
  19. P.R.B. Devloo, S.M. Gomes, T. Quinelato and S. Tian, Enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry. Comput. Math. App. 79 (2020) 2678–2700. [Google Scholar]
  20. O. Durán, P.R.B. Devloo, S.M. Gomes and F. Valentin, A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers. Comput. Meth. Appl. Mech. Eng. 354 (2019) 213–244. [Google Scholar]
  21. Y. Efendiev and Y. Hou, Multiscale Finite Element Methods: Theory and Applications, Tutorials in the Applied Mathematical Sciences. Springer, New York 4 (2009). [Google Scholar]
  22. R.S. Falk, Finite element methods for linear elasticity. In: Mixed Finite Elements, Compatibility Conditions, and Applications, edited by D. Boffi and L. Gastaldi. Vol. 1939 of Lecture Notes in MathematicsSpringer, Berlin, Heidelberg (2008) 159–194. [Google Scholar]
  23. M. Farhloul and M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math. 76 (1997) 419–440. [Google Scholar]
  24. V. Girault and P.A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer-Verlag, New York (1991). [Google Scholar]
  25. A.T.A. Gomes, D. Paredes, W.D.S. Pereira, R.P. Souto and F. Valentin, Performance analysis of the MHM simulator in a petascale machine. In: Proceedings of the XXXVIII Iberian Latin American Congress on Computational Methods in Engineering. ABMEC (2017). [Google Scholar]
  26. C. Harder, D. Paredes and F. Valentin, A family of multiscale hybrid-mixed finite element methods for the darcy equation with rough coefficients. J. Comput. Phys. 245 (2013) 107–130. [Google Scholar]
  27. C. Harder, D. Paredes and F. Valentin, On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogenous coefficients. SIAM Multiscale Model. Simul. 3 (2015) 491–518. [Google Scholar]
  28. C. Harder, A.L. Madureira and F. Valentin, A hybrid-mixed method for elasticity. ESAIM:M2AN 50 (2016) 311–336. [EDP Sciences] [Google Scholar]
  29. P. Henning and A. Persson, A multiscale method for linear elasticity reducing poisson locking. Comput. Meth. Appl. Mech. Eng. 310 (2016) 156–171. [Google Scholar]
  30. E. Khattatov and I. Yotov, Domain decomposition and multiscale mortar mixed finite elements methods for linear elasticity with weak stress symmetry. Math. Model. Numer. Anal. 53 (2019) 2081–2108. [Google Scholar]
  31. M. Kuchta, K.-A. Mardal and M. Mortensen, On the singular Neumann problem in linear elasticity. Numer. Linear. Algebra Appl. 26 (2019) e2212. [Google Scholar]
  32. A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems. Math. Comp. 83 (2014) 2583–2603. [Google Scholar]
  33. L. Mansfield, Finite element subspaces with optimal rates of convergence for the stationary Stokes problem. RAIRO Anal. Numer. 16 (1982) 49–66. [Google Scholar]
  34. D. Paredes, F. Valentin and H.M. Versieux, On the robustness of multiscale hybrid-mixed methods. Math. Comput. 86 (2016) 525–548. [Google Scholar]
  35. W.S. Pereira, Multiscale hybrid-mixed methods for heterogeneous elastic models. Ph.D. thesis, LNCC, RJ, BR (2019). [Google Scholar]
  36. W. Pereira and F. Valentin, A locking-free MHM method for elasticity. In: Vol. 5 of Proceeding Series of the Brazilian Society of Computational and Applied Mathematics (2017). [Google Scholar]
  37. P.A. Raviart and J.M. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp. 31 (1997) 391–413. [Google Scholar]
  38. J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods. In: Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science Publishers (1991) 527–639. [Google Scholar]
  39. D. Siqueira, P.R.B. Devloo and S.M. Gomes, A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces. J. Comput. Appl. Math. 240 (2013) 204–214. [Google Scholar]
  40. R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513–538. [Google Scholar]

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